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Great textbook, I learned all the topology I know from it. Previously, Category Theory was taught as a field that connects branches of math, and thus in terms o

by lgdw 4y ago

Great textbook, I learned all the topology I know from it. Previously, Category Theory was taught as a field that connects branches of math, and thus in terms of other concepts. But recently there's a movement to view Category Theory as the definitive underlying field of math (instead of set theory), and teach different fields of math in terms of Category Theory rather than vice versa (a new-new math in a sense). I learned Category Theory well before learning abstract algebra and topology, and the embedding of Topology in Category Theory was seamless and intuitive; I feel as though this book proves that this new CT-centric view of math education has merit. One of the authors, Tai-Danae Bradley, also runs math3ma [1] and is a prominent figure in Applied Category Theory. I had the pleasure of hearing her talk, and her way of explaining abstractions is very easy to understand despite Category Theory being fairly obtuse at times (looking at you, Mac Lane!) Also, an obligatory shilling of the Topos Institute [2]. They're a research institution based in Berkeley, and they have weekly talks on Category Theory that they release on youtube. If you're interested in the categorification of mathematics, you need to check them out. [1] https://www.math3ma.com/ https://www.math3ma.com/ [2] https://topos.site/ https://topos.site/

bwestergard 4y ago
"I learned Category Theory well before learning abstract algebra and topology, and the embedding of Topology in Category Theory was seamless and intuitive; I feel as though this book proves that this new CT-centric view of math education has merit." Could you tell us a bit more about your educational history and motivations for studying these topics in this order?
- lgdw 4y ago
  While I was in highschool I attended a lecture by David Spivak on a whim and was fascinated by the field ever since. Before really discovering Category Theory, I was more interested in low-level computer architecture and design (although I'm not very knowledgeable by any means) so I didn't really encounter Category Theory through the means that most Computer Science people do (FP, Haskell, etc). Once I learned Category Theory I became more interested in other fields of math.
  - realitygrill 4y ago
    If one was a young or returning adult with interest but little background in mathematics, would you suggest learning category theory first (assuming the adult is interested)? How would you suggest going about doing this as a path?
    - lgdw 4y ago
      Although it would depend on what kind of fields of math you are interested in (algebra, analysis, topology, etc), I think that you can't really _go wrong_ per se by learning Category Theory first, even if many of the examples/uses of category theory won't make sense at first. Of course, learning Category Theory first is definitely unorthodox; at least in college, you usually first learn basic algebra (Abstract Algebra, Linear Algebra) along with analysis (Complex Analysis, Real Analysis) and "advanced calculus" (Differential Equations, Multivariable Calculus). Fields like Category Theory usually come after that and are taught mostly in grad school, but at its core learning Category Theory doesn't require knowing a lot of prerequisites so I think in terms of accessibility it resides alongside fields like Linear Algebra or Group Theory. An advantage of learning Category Theory first is that once you have a decent grasp of it, you'd have the mathematical vocabulary to describe concepts learned in different fields; a homotopy is a 2-morphism in the category of topological spaces, for example. That being said, if you like algebra the most, learn algebra first. If you enjoy topology, learn topology. There really isn't a "right place" to start with mathematics, and as long as you avoid fields of math that build heavily on other fields of math like K-theory or representation theory, you'll have a decent starting background in math. Most fields of math, not just Category Theory, have analogues to other fields (and Category Theory acts only to really formalize this connection), so you can't really go wrong with starting with something like Linear Algebra or Group Theory.
lgdw 4y ago
I forgot to add that Algebra Chapter 0 takes a similar (albeit slightly different) approach of teaching Abstract Algebra in terms of Category Theory. I don't have a link right now but I'm sure you can find it on libgen. (I've only read the first few chapters of Algebra Chapter 0 yet, but from what I've heard the rest of the textbook is as good as the first bit.)
ruuda 4y ago
> the embedding of Topology in Category Theory was seamless and intuitive It is no surprise that it is a good fit, category theory first emerged out of topology.
bigbillheck 4y ago
> there's a movement to view Category Theory as the definitive underlying field of math Category theory doesn't have much to say about most of analysis.
- kisonecat 4y ago
  There's definitely some connections between analysis and category theory though! There are "self-adjoint operators" in analysis and there are "adjoint functors" in category theory, and it is sort of fun to think about the analogy there.
fn-mote 4y ago
> Great textbook, I learned all the topology I know from it. The kind of topology this book focuses on is what I learned as "point-set topology". I guess the parent considers topology to not contain algebraic topology, since they go on to say they know more than this book contains. You could turn this statement around - what if you learned all of the category theory you knew out of this book? Maybe someone can comment. The start of the book and their take on point-set topology seems reasonable. I saw all of the familiar theorems without any eye-popping machinery being used. I was very happy to see some technically important items in the book (the compact-open topology, and the corresponding topology on Hom). I would be very interested in a semi-experienced reader's take on Chapter 5. [1] If you don't already know category theory, is this a reasonable introduction? (Is the on ramp ever smooth?) I can't vouch for what kind of intro to (point-set) topology the book gives, but I can certainly say that I would prefer this book over MacLane's "Categories for the Working Mathematician" as an introduction to category theory, for what it's worth. [1] https://assets.pubpub.org/6d1dqgg9/51597355090422.pdf https://assets.pubpub.org/6d1dqgg9/51597355090422.pdf
- lgdw 4y ago
  Yes, this textbook is a great introduction to point-set topology. I'm not very familiar with algebraic topology so I can't comment on anything regarding that. I did learn a fair bit of category theory that I didn't know prior, like a formal explanation of deriving the Yoneda Lemma (I read this book before I read CWM). I think it's definitely possible to learn category theory from this book, especially if you already have a strong intuition for Topology.
daxfohl 4y ago
What does it mean, that it's replacing set theory? Is it more fundamental than set theory, in the sense that the latter can be defined in terms of it? Or is it different from set theory entirely, meaning different math comes from each?
- Twisol 4y ago
  Category theory might be easier to think about as a different way to think about the same objects we've always studied in mathematics. Since it's a formal theory (just like set theory), we can also study category theory mathematically. In fact, we can study set theory categorically and we can study categories set-theoretically -- they aren't really at odds with each other. If you wanted to study set theory from a categorical perspective, you would likely be interested in topos theory, which captures essential properties of set theory and generalizes them to a variety of interesting settings. There's also a really good (and IMHO approachable) paper by Tom Leinster [1] on how to see set theory from a categorical perspective, without going all the way out to toposes. I think it's a good way to understand what category theorists try to emphasize. If you wanted to study categories from a set-theoretic perspective, you're in luck -- that's probably the most common approach to learning category theory in the first place! Studying categories with categories is also possible, but you already need to be fairly comfortable with categories in order to model them internally. (Studying set theory with set theory is much the same -- models of ZFC can get wild.) (You can also learn category theory from first principles, without thinking in terms of sets. This isn't really any different from set theory, either -- set theory is often peoples' first introductions to abstract mathematics at the undergraduate level, so there isn't really much choice in the matter. For my money, though, I think category theory from first principles is a little less weird.) [1]: https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543
- i_k_k 4y ago
  I never understood this either... Category theory works with collections, which can be modeled as sets -- or if you want to deal with categories of sets and other "large" categories -- as proper classes. And you can use a set theory that incorporates classes, like Morse-Kelley, to model this. IOW, it seems like category theory can be modeled in set theory. On the other hand, I've not seen a rigorous set of axioms for category theory that didn't presuppose a notion of set or category or "collection". Set theory also sheds a lot of light on what feel like foundational issues: sizes of sets, independence of axioms, etc.; I haven't seen something similar out of category theory. Love for a category theory partisan to chime in with more here -- I'm definitely not an expert.
orangea 4y ago
Why are you capitalizing "category theory" and not "topology", "algebra", or "math"?
- lgdw 4y ago
  Sorry, force of habit.
flowingfocus 4y ago
Thanks for the recommendations! > Great textbook, I learned all the topology I know from it. Hatcher's Algebraic Topology was that book for me. https://pi.math.cornell.edu/~hatcher/AT/ATpage.html https://pi.math.cornell.edu/~hatcher/AT/ATpage.html
data_maan 4y ago
> But recently there's a movement to view Category Theory as the definitive underlying field of math Spoken like a true algebraist. For anything related to stats, PDEs, and optimization (basically that subset if mathematics that is most useful to other sciences), category theory is a horrible foundation. While it seems you can recover (with a looot of work) some existing theory, no sane researcher in these fields uses category theory. And there is also no motivation to do so, since unlike for algebraic topology, algebra etc. the categorical viewpoint doesn't really make it clearer, what, e.g., the weak solution to a PDEs is, and why it coincidences (or not) with the classical solution.
- maxiepoo 4y ago
  I think OP is really overstating things. "the definitive underlying field of math replacing set theory" is something Lawvere was trying in the 60s and it never really took off. Category theory isn't really a replacement for set theory any more than order theory is a replacement for set theory. What is important is not what your "foundational system" is, but that just like there are many concepts we learn in the context of set theory but are just very generally useful (equivalence classes, cardinality, injective/surjective functions), there are also similarly very useful concepts in category theory like adjoint functors, limits and universal mapping properties.
- q-big 4y ago
  > For anything related to stats, PDEs, and optimization (basically that subset if mathematics that is most useful to other sciences), category theory is a horrible foundation. For analysis, Peter Scholze and Dustin Clausen would disagree: > https://en.wikipedia.org/wiki/Condensed_mathematics https://en.wikipedia.org/wiki/Condensed_mathematics To quote from https://www.math.uni-bonn.de/people/scholze/Analytic.pdf https://www.math.uni-bonn.de/people/scholze/Analytic.pdf linked there: "Mumford writes in Curves and their Jacobians: “[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.” For some reason, this secret plot has so far stopped short of taking over analysis. The goal of this course is to launch a new attack, turning functional analysis into a branch of commutative algebra, and various types of analytic geometry (like manifolds) into algebraic geometry. Whether this will make these subjects equally esoteric will be left to the reader’s judgement. [...] The author always had the impression that the highly categorical techniques of algebraic geometry could not possibly be applied in analytic situations; and certainly not over the real numbers. The goal of this course is to correct this impression."

by lgdw 4y ago

bwestergard 4y ago

"I learned Category Theory well before learning abstract algebra and topology, and the embedding of Topology in Category Theory was seamless and intuitive; I feel as though this book proves that this new CT-centric view of math education has merit." Could you tell us a bit more about your educational history and motivations for studying these topics in this order?

lgdw 4y ago
While I was in highschool I attended a lecture by David Spivak on a whim and was fascinated by the field ever since. Before really discovering Category Theory, I was more interested in low-level computer architecture and design (although I'm not very knowledgeable by any means) so I didn't really encounter Category Theory through the means that most Computer Science people do (FP, Haskell, etc). Once I learned Category Theory I became more interested in other fields of math.
- realitygrill 4y ago
  If one was a young or returning adult with interest but little background in mathematics, would you suggest learning category theory first (assuming the adult is interested)? How would you suggest going about doing this as a path?
  - lgdw 4y ago
    Although it would depend on what kind of fields of math you are interested in (algebra, analysis, topology, etc), I think that you can't really _go wrong_ per se by learning Category Theory first, even if many of the examples/uses of category theory won't make sense at first. Of course, learning Category Theory first is definitely unorthodox; at least in college, you usually first learn basic algebra (Abstract Algebra, Linear Algebra) along with analysis (Complex Analysis, Real Analysis) and "advanced calculus" (Differential Equations, Multivariable Calculus). Fields like Category Theory usually come after that and are taught mostly in grad school, but at its core learning Category Theory doesn't require knowing a lot of prerequisites so I think in terms of accessibility it resides alongside fields like Linear Algebra or Group Theory. An advantage of learning Category Theory first is that once you have a decent grasp of it, you'd have the mathematical vocabulary to describe concepts learned in different fields; a homotopy is a 2-morphism in the category of topological spaces, for example. That being said, if you like algebra the most, learn algebra first. If you enjoy topology, learn topology. There really isn't a "right place" to start with mathematics, and as long as you avoid fields of math that build heavily on other fields of math like K-theory or representation theory, you'll have a decent starting background in math. Most fields of math, not just Category Theory, have analogues to other fields (and Category Theory acts only to really formalize this connection), so you can't really go wrong with starting with something like Linear Algebra or Group Theory.

lgdw 4y ago

I forgot to add that Algebra Chapter 0 takes a similar (albeit slightly different) approach of teaching Abstract Algebra in terms of Category Theory. I don't have a link right now but I'm sure you can find it on libgen. (I've only read the first few chapters of Algebra Chapter 0 yet, but from what I've heard the rest of the textbook is as good as the first bit.)

ruuda 4y ago

> the embedding of Topology in Category Theory was seamless and intuitive It is no surprise that it is a good fit, category theory first emerged out of topology.

bigbillheck 4y ago

> there's a movement to view Category Theory as the definitive underlying field of math Category theory doesn't have much to say about most of analysis.

kisonecat 4y ago
There's definitely some connections between analysis and category theory though! There are "self-adjoint operators" in analysis and there are "adjoint functors" in category theory, and it is sort of fun to think about the analogy there.

fn-mote 4y ago

> Great textbook, I learned all the topology I know from it. The kind of topology this book focuses on is what I learned as "point-set topology". I guess the parent considers topology to not contain algebraic topology, since they go on to say they know more than this book contains. You could turn this statement around - what if you learned all of the category theory you knew out of this book? Maybe someone can comment. The start of the book and their take on point-set topology seems reasonable. I saw all of the familiar theorems without any eye-popping machinery being used. I was very happy to see some technically important items in the book (the compact-open topology, and the corresponding topology on Hom). I would be very interested in a semi-experienced reader's take on Chapter 5. [1] If you don't already know category theory, is this a reasonable introduction? (Is the on ramp ever smooth?) I can't vouch for what kind of intro to (point-set) topology the book gives, but I can certainly say that I would prefer this book over MacLane's "Categories for the Working Mathematician" as an introduction to category theory, for what it's worth. [1] https://assets.pubpub.org/6d1dqgg9/51597355090422.pdf https://assets.pubpub.org/6d1dqgg9/51597355090422.pdf

lgdw 4y ago
Yes, this textbook is a great introduction to point-set topology. I'm not very familiar with algebraic topology so I can't comment on anything regarding that. I did learn a fair bit of category theory that I didn't know prior, like a formal explanation of deriving the Yoneda Lemma (I read this book before I read CWM). I think it's definitely possible to learn category theory from this book, especially if you already have a strong intuition for Topology.

daxfohl 4y ago

What does it mean, that it's replacing set theory? Is it more fundamental than set theory, in the sense that the latter can be defined in terms of it? Or is it different from set theory entirely, meaning different math comes from each?

Twisol 4y ago
Category theory might be easier to think about as a different way to think about the same objects we've always studied in mathematics. Since it's a formal theory (just like set theory), we can also study category theory mathematically. In fact, we can study set theory categorically and we can study categories set-theoretically -- they aren't really at odds with each other. If you wanted to study set theory from a categorical perspective, you would likely be interested in topos theory, which captures essential properties of set theory and generalizes them to a variety of interesting settings. There's also a really good (and IMHO approachable) paper by Tom Leinster [1] on how to see set theory from a categorical perspective, without going all the way out to toposes. I think it's a good way to understand what category theorists try to emphasize. If you wanted to study categories from a set-theoretic perspective, you're in luck -- that's probably the most common approach to learning category theory in the first place! Studying categories with categories is also possible, but you already need to be fairly comfortable with categories in order to model them internally. (Studying set theory with set theory is much the same -- models of ZFC can get wild.) (You can also learn category theory from first principles, without thinking in terms of sets. This isn't really any different from set theory, either -- set theory is often peoples' first introductions to abstract mathematics at the undergraduate level, so there isn't really much choice in the matter. For my money, though, I think category theory from first principles is a little less weird.) [1]: https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543
i_k_k 4y ago
I never understood this either... Category theory works with collections, which can be modeled as sets -- or if you want to deal with categories of sets and other "large" categories -- as proper classes. And you can use a set theory that incorporates classes, like Morse-Kelley, to model this. IOW, it seems like category theory can be modeled in set theory. On the other hand, I've not seen a rigorous set of axioms for category theory that didn't presuppose a notion of set or category or "collection". Set theory also sheds a lot of light on what feel like foundational issues: sizes of sets, independence of axioms, etc.; I haven't seen something similar out of category theory. Love for a category theory partisan to chime in with more here -- I'm definitely not an expert.

orangea 4y ago

Why are you capitalizing "category theory" and not "topology", "algebra", or "math"?

lgdw 4y ago
Sorry, force of habit.

flowingfocus 4y ago

Thanks for the recommendations! > Great textbook, I learned all the topology I know from it. Hatcher's Algebraic Topology was that book for me. https://pi.math.cornell.edu/~hatcher/AT/ATpage.html https://pi.math.cornell.edu/~hatcher/AT/ATpage.html

data_maan 4y ago

> But recently there's a movement to view Category Theory as the definitive underlying field of math Spoken like a true algebraist. For anything related to stats, PDEs, and optimization (basically that subset if mathematics that is most useful to other sciences), category theory is a horrible foundation. While it seems you can recover (with a looot of work) some existing theory, no sane researcher in these fields uses category theory. And there is also no motivation to do so, since unlike for algebraic topology, algebra etc. the categorical viewpoint doesn't really make it clearer, what, e.g., the weak solution to a PDEs is, and why it coincidences (or not) with the classical solution.

maxiepoo 4y ago
I think OP is really overstating things. "the definitive underlying field of math replacing set theory" is something Lawvere was trying in the 60s and it never really took off. Category theory isn't really a replacement for set theory any more than order theory is a replacement for set theory. What is important is not what your "foundational system" is, but that just like there are many concepts we learn in the context of set theory but are just very generally useful (equivalence classes, cardinality, injective/surjective functions), there are also similarly very useful concepts in category theory like adjoint functors, limits and universal mapping properties.
q-big 4y ago
> For anything related to stats, PDEs, and optimization (basically that subset if mathematics that is most useful to other sciences), category theory is a horrible foundation. For analysis, Peter Scholze and Dustin Clausen would disagree: > https://en.wikipedia.org/wiki/Condensed_mathematics https://en.wikipedia.org/wiki/Condensed_mathematics To quote from https://www.math.uni-bonn.de/people/scholze/Analytic.pdf https://www.math.uni-bonn.de/people/scholze/Analytic.pdf linked there: "Mumford writes in Curves and their Jacobians: “[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.” For some reason, this secret plot has so far stopped short of taking over analysis. The goal of this course is to launch a new attack, turning functional analysis into a branch of commutative algebra, and various types of analytic geometry (like manifolds) into algebraic geometry. Whether this will make these subjects equally esoteric will be left to the reader’s judgement. [...] The author always had the impression that the highly categorical techniques of algebraic geometry could not possibly be applied in analytic situations; and certainly not over the real numbers. The goal of this course is to correct this impression."